Almost Sure Entropy and the Variational Principle for Random Fields with Unbounded State Space
图像的分割和配准中英文翻译
外文文献资料翻译：李睿钦指导老师：刘文军Medical image registration with partial dataSenthil Periaswamy，Hany FaridThe goal of image registration is to find a transformation that aligns one image to another. Medical image registration has emerged from this broad area of research as a particularly active field. This activity is due in part to the many clinical applications including diagnosis, longitudinal studies, and surgical planning, and to the need for registration across different imaging modalities (e.g., MRI, CT, PET, X-ray, etc.). Medical image registration, however, still presents many challenges. Several notable difficulties are (1) the transformation between images can vary widely and be highly non-rigid in nature; (2) images acquired from different modalities may differ significantly in overall appearance and resolution; (3) there may not be a one-to-one correspondence between the images (missing/partial data); and (4) each imaging modality introduces its own unique challenges, making it difficult to develop a single generic registration algorithm.In estimating the transformation that aligns two images we must choose: (1) to estimate the transformation between a small number of extracted features, or between the complete unprocessed intensity images; (2) a model that describes the geometric transformation; (3) whether to and how to explicitly model intensity changes; (4) an error metric that incorporates the previous three choices; and (5) a minimization technique for minimizing the error metric, yielding the desired transformation.Feature-based approaches extract a (typically small) number of corresponding landmarks or features between the pair of images to be registered. The overall transformation is estimated from these features. Common features include corresponding points, edges, contours or surfaces. These features may be specified manually or extracted automatically. Fiducial markers may also be used as features;these markers are usually selected to be visible in different modalities. Feature-based approaches have the advantage of greatly reducing computational complexity. Depending on the feature extraction process, these approaches may also be more robust to intensity variations that arise during, for example, cross modality registration. Also, features may be chosen to help reduce sensor noise. These approaches can be, however, highly sensitive to the accuracy of the feature extraction. Intensity-based approaches, on the other hand, estimate the transformation between the entire intensity images. Such an approach is typically more computationally demanding, but avoids the difficulties of a feature extraction stage.Independent of the choice of a feature- or intensity-based technique, a model describing the geometric transform is required. A common and straightforward choice is a model that embodies a single global transformation. The problem of estimating a global translation and rotation parameter has been studied in detail, and a closed form solution was proposed by Schonemann. Other closed-form solutions include methods based on singular value decomposition (SVD), eigenvalue-eigenvector decomposition and unit quaternions. One idea for a global transformation model is to use polynomials. For example, a zeroth-order polynomial limits the transformation to simple translations, a first-order polynomial allows for an affine transformation, and, of course, higher-order polynomials can be employed yielding progressively more flexible transformations. For example, the registration package Automated Image Registration (AIR) can employ (as an option) a fifth-order polynomial consisting of 168 parameters (for 3-D registration). The global approach has the advantage that the model consists of a relatively small number of parameters to be estimated, and the global nature of the model ensures a consistent transformation across the entire image. The disadvantage of this approach is that estimation of higher-order polynomials can lead to an unstable transformation, especially near the image boundaries. In addition, a relatively small and local perturbation can cause disproportionate and unpredictable changes in the overall transformation. An alternative to these global approaches are techniques that model the global transformation as a piecewise collection of local transformations. For example, the transformation between each local region may bemodeled with a low-order polynomial, and global consistency is enforced via some form of a smoothness constraint. The advantage of such an approach is that it is capable of modeling highly nonlinear transformations without the numerical instability of high-order global models. The disadvantage is one of computational inefficiency due to the significantly larger number of model parameters that need to be estimated, and the need to guarantee global consistency. Low-order polynomials are, of course, only one of many possible local models that may be employed. Other local models include B-splines, thin-plate splines, and a multitude of related techniques. The package Statistical Parametric Mapping (SPM) uses the low-frequency discrete cosine basis functions, where a bending-energy function is used to ensure global consistency. Physics-based techniques that compute a local geometric transform include those based on the Navier–Stokes equilibrium equations for linear elastici and those based on viscous fluid approaches.Under certain conditions a purely geometric transformation is sufficient to model the transformation between a pair of images. Under many real-world conditions, however, the images undergo changes in both geometry and intensity (e.g., brightness and contrast). Many registration techniques attempt to remove these intensity differences with a pre-processing stage, such as histogram matching or homomorphic filtering. The issues involved with modeling intensity differences are similar to those involved in choosing a geometric model. Because the simultaneous estimation of geometric and intensity changes can be difficult, few techniques build explicit models of intensity differences. A few notable exceptions include AIR, in which global intensity differences are modeled with a single multiplicative contrast term, and SPM in which local intensity differences are modeled with a basis function approach.Having decided upon a transformation model, the task of estimating the model parameters begins. As a first step, an error function in the model parameters must be chosen. This error function should embody some notion of what is meant for a pair of images to be registered. Perhaps the most common choice is a mean square error (MSE), defined as the mean of the square of the differences (in either feature distance or intensity) between the pair of images. This metric is easy to compute and oftenaffords simple minimization techniques. A variation of this metric is the unnormalized correlation coefficient applicable to intensity-based techniques. This error metric is defined as the sum of the point-wise products of the image intensities, and can be efficiently computed using Fourier techniques. A disadvantage of these error metrics is that images that would qualitatively be considered to be in good registration may still have large errors due to, for example, intensity variations, or slight misalignments. Another error metric (included in AIR) is the ratio of image uniformity (RIU) defined as the normalized standard deviation of the ratio of image intensities. Such a metric is invariant to overall intensity scale differences, but typically leads to nonlinear minimization schemes. Mutual information, entropy and the Pearson product moment cross correlation are just a few examples of other possible error functions. Such error metrics are often adopted to deal with the lack of an explicit model of intensity transformations .In the final step of registration, the chosen error function is minimized yielding the desired model parameters. In the most straightforward case, least-squares estimation is used when the error function is linear in the unknown model parameters. This closed-form solution is attractive as it avoids the pitfalls of iterative minimization schemes such as gradient-descent or simulated annealing. Such nonlinear minimization schemes are, however, necessary due to an often nonlinear error function. A reasonable compromise between these approaches is to begin with a linear error function, solve using least-squares, and use this solution as a starting point for a nonlinear minimization.译文：部分信息的医学图像配准Senthil Periaswamy，Hany Farid图像配准的目的是找到一种能把一副图像对准另外一副图像的变换算法。
LPFTVD_2014_Feb04
However, the signals arising in some applications are more complex: they are neither isolated to a specific frequency band nor do they admit a highly sparse representation. For such signals, neither LTI filtering nor sparsity-based denoising is appropriate by itself. Can conventional LTI filtering and more recent sparsity-based denoising methods be combined in a principled way, to effectively filter (denoise) a wider class of signals than either approach can alone? This paper addresses the problem of filtering noisy data where the underlying signal comprises a low-frequency component and a sparse or sparse-derivative component. It is assumed here that the noisy data y (n) can be modeled as y (n) = f (n) + x(n) + w(n), n = 0, . . . , N − 1 (1)
Abstract—This paper seeks to combine linear time-invariant (LTI) filtering and sparsity-based denoising in a principled way in order to effectively filter (denoise) a wider class of signals. LTI filtering is most suitable for signals restricted to a known frequency band, while sparsity-based denoising is suitable for signals admitting a sparse representation with respect to a known transform. However, some signals cannot be accurately categorized as either band-limited or sparse. This paper addresses the problem of filtering noisy data for the particular case where the underlying signal comprises a low-frequency component and a sparse or sparse-derivative component. A convex optimization approach is presented and two algorithms derived, one based on majorization-minimization (MM), the other based on the alternating direction method of multipliers (ADMM). It is shown that a particular choice of discrete-time filter, namely zero-phase non-causal recursive filters for finite-length data formulated in teห้องสมุดไป่ตู้ms of banded matrices, makes the algorithms effective and computationally efficient. The efficiency stems from the use of fast algorithms for solving banded systems of linear equations. The method is illustrated using data from a physiologicalmeasurement technique (i.e., near infrared spectroscopic time series imaging) that in many cases yields data that is wellapproximated as the sum of low-frequency, sparse or sparsederivative, and noise components.
Almost Sure Convergence Theorems of Rate of Coin Tosses for Random Number Generation by Int
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Interval algorithm for general sources 1) Set l = m = 1; s = t = (null string), s = t = 0; s = t = 1, I (s) = [ s ; s ) and J (t) = [ t; t ). 2) Partition the interval J (t) = [ t ; t) into M disjoint subintervals J (t1); J (t2); 1 1 1 ; J (tM ) such that J (tj ) = [ tj ; tj ) (j = 1; 2; 1 1 1 ; M ) where tj = t + (t 0 t )Qj 01 tj = t + (t 0 t )Qj
基于相对熵和esd检测的视频关键帧抽取算法
摘要随着互联网以及多媒体技术的飞速发展，使得数字视频在人们的日常生活中越来越普及。
人们可以方便的使用手机等便携设备拍摄数字视频，在线视频播放网站如雨后春笋般涌现，大型视频数据库也愈发常见。
如何高效的存储和管理大量的视频内容信息成为亟待解决的问题。
并且，伴随着视频内容的丰富化和视频种类的多样化，人们迫切需要一种快速有效的了解视频内容信息的方式。
然而要实现对视频数据的理解和分析，就需要完成大量的视频数据处理，这在实际应用中不是一项容易的工作。
抽取视频序列中的关键帧序列能够很好的解决上述要求，即通过一组具有代表性的视频帧序列——视频关键帧，来表示原始视频序列的主要内容信息。
本文首先对视频关键帧抽取的相关知识做了概要介绍。
在这个基础上，本文提出了一种新的视频关键帧抽取方法。
本方法首先计算视频相邻帧之间的相对熵（Relative Entropy,简称RE）或相对熵的平方根（Square Root of Relative Entropy,简称SRRE）来表示视频相邻帧之间的差异值，然后通过统计学中的离群值检测算法——极值学生化离差（Extreme Studentized Deviate，简称ESD）检测法寻找离群值，再通过多项式回归的方式进行修正，寻找最优分割阈值定位镜头边界，实现视频序列的自适应镜头分割。
为了进一步分析视频每个镜头的内容信息，在此基础上本文根据镜头内容变化的剧烈程度将镜头进行细分为不同类型的子镜头，并在每个子镜头内部抽取关键帧。
另外，本文还提出一种采用层次策略的视频关键帧的多尺度摘要方案。
通过大量视频数据的实验测试，将本文中提出的方法的关键帧结果与其它方法的关键帧结果进行对比，本文方法无论是在客观评价还是在主观评价方面都优于对比的方法，而且本文方法基本达到了普适性和实时性的效果。
关键词：关键帧抽取，相对熵，ESD检测，多尺度ABSTRACTWith the rapid development of Internet and multimedia technology, the digital videos have become more and more popular in people's daily life. People can easily use mobile phones and other portable devices to shoot digital videos; numerous online video playback sites have sprung up; large video databases are increasingly common in life. How to effectively store and manage a large amount of video content information becomes an urgent problem to be solved. In addition, with the richness and the various varieties of the video content, people urgently need a fast and effective way to understand the information that videos carry. However, to better understand and analysis the videos, it is inevitable to do much video data processing work, which is not easy in the practical applications. Extracting the key frame sequences of the video sequences can solve the problem above well, that is, the main content information of the original video sequences is represented by a set of representative video frame sequences.In this thesis, we first introduce the relevant knowledge of video key frame extraction. Based on this, a new method to effectively extract video key frames is presented. We first calculate the relative entropy(RE) or the square root of relative entropy(SRRE) between adjacent frames of the video, as the difference between adjacent frames. Then the statistical outlier detection algorithm Studentized Deviate Extreme (ESD) is utilized to identify outliers. Then we find the optimal segmentation threshold to locate the shot boundary through the method of polynomial regression, and implement the adaptive shot segmentation of video sequences. In order to further analyze the content information in each video shot, we subdivide the video shots into different types, according to the extent of variation of the shot content, then extract key frames in each sub shot. In addition, this thesis proposes a multi scale abstract scheme for video key frames based on the hierarchical strategy. Extensive experimental results of a large amount of video data indicate that, compared with other methods, the proposed method has better performance, in terms of both the objective and subjective evaluation. Meanwhile, the new method achieves the basic universality and real-time effect.KEY WORDS:Key frame extraction, Relative entropy, ESD test, Multiscale目录摘要 (I)ABSTRACT (II)第1章绪论 (1)1.1研究背景与意义 (1)1.2研究现状 (2)1.3本文创新 (4)1.4论文结构 (5)第2章视频关键帧抽取相关介绍 (6)2.1视频数据的特征 (6)2.2视频序列的结构 (8)2.3视频镜头变换类型 (9)2.4视频图像特征 (10)第3章关键帧抽取方法以及多尺度关键帧摘要 (12)3.1关键帧抽取方法 (12)3.1.1帧间距离度量 (13)3.1.2分割视频镜头 (15)3.1.3分割视频子镜头 (18)3.1.4视频关键帧抽取 (19)3.2多尺度关键帧摘要 (20)3.2.1获取多尺度关键帧摘要策略 (20)3.2.2多尺度关键帧摘要结果展示 (22)3.3小结 (23)第4章实验结果 (24)4.1客观评价 (25)4.1.1 VSE视频抽样错误率 (26)4.1.2 FID保真度 (27)4.1.3客观评价的结果 (27)4.2主观评价 (31)4.2.1主观评价的结果 (31)4.2.2视频关键帧抽取结果分析 (33)4.3时间空间统计 (34)4.4小结 (38)第5章总结与展望 (39)5.1总结 (39)5.2展望 (40)参考文献 (41)发表论文和参加科研情况说明 (44)致谢 (45)第1章绪论1.1研究背景与意义随着便携式数字产品的快速推广和高效视频压缩技术的不断进步，以及计算机网络技术的发展和网络传输速度的提升，人们能够方便的使用便携式设备录制视频，并且可以发布到互联网上与他人共同分享。
Strategic Entry Deterrence and the Behavior of
Strategic Entry Deterrence and the Behavior of Pharmaceutical Incumbents Prior to Patent Expiration1Glenn EllisonMIT and NBERandSara Fisher EllisonMITApril20071email:gellison@,sellison@.This work was supported by the National Science Foundation(grants SES-9818534,SES-0219205,and SES-0550897),the Sloan Foundation,and the Hoover Institute.We thank Emek Basker,Tom Chang,David Hwang,and Alan Sorensen for excellent research assistance,Gillian Currie for help in constructing the data set,Katie MacFarlane for guidance on institutional details,and Ditas Riad and Rhea Mihalison for their assistance on data issues.We also thank Fiona Scott Morton for her part in our joint data collection efforts and for providing other data to us.Finally,we thank Marcus Asplund,Ernie Berndt,Steve Berry,Richard Blundell,Judy Chevalier,Paul Joskow,Nancy Rose,Otto Toivanen,and seminar participants at various schools for helpful comments.AbstractThis paper develops a new approach to testing for strategic entry deterrence and applies it to the behavior of pharmaceutical incumbents just before they lose patent protection. The approach involves looking at a cross-section of markets and examining whether behav-ior is nonmonotonic in the size of the market.Under certain conditions,investment levels will be monotone in market size iffirms are not influenced by a desire to deter entry.Strate-gic investments,however,may be nonmonotone because entry deterrence is unnecessary in very small markets and impossible in very large ones,resulting in overall nonmonotonic investment.The pharmaceutical data contain advertising,product proliferation,and pric-ing information for a sample of drugs which lost patent protection between1986and1992. Among thefindings consistent with an entry deterrence motivation are that incumbents in markets of intermediate size have lower levels of advertising and are more likely to reduce advertising immediately prior to patent expiration.Glenn Ellison Sara Fisher EllisonDepartment of Economics Department of EconomicsMassachusetts Institute of Technology Massachusetts Institute of Technology50Memorial Drive50Memorial DriveCambridge,MA02142-1347Cambridge,MA02142-13471IntroductionThe insight thatfirms may make“strategic investments”to alter future competitive con-ditions is one of the most fundamental ideas in industrial organization.The chapter on “Entry,Accomodation,and Exit”is easily the longest in Tirole’s(1988)text.In it,Tirole reviews arguments about how excess capacity,capital structure,advertising,contractual practices,learning-by-doing,and long-run decisions can be used to deter entry.1Strategic investment models are difficult to test directly,however,and the vast majority of this lit-erature is theoretical.In this paper,we propose a new empirical approach for examining strategic entry deterrence.Our applied focus is on the pharmaceutical ing a panel of drugs that lost their U.S.patent protection between1986and1992,we explore how pharmaceutical incumbents have dealt with the threat of generic entry.We examine incumbents’advertis-ing,product proliferation,and pricing decisions as patent expiration approached,and ask whether the behaviors appear to be influenced by an entry-deterrence motive.We begin in Section2with a discussion of strategic entry deterrence and some moti-vation for our approach.We modify the textbook model to bring it closer to empirical applications:we assume that entry costs are random and unknown to the incumbent so that it is impossible to perfectly forecast whether entry will occur.We review what is meant by“strategic entry deterrence”in this setting.We note that the incentive to deter entry will be stronger in intermediate-sized markets than in very small or very large markets. In the former,no investments are needed to deter entry.In the latter,deterring entry is often impossible.A simple numerical example illustrates how the nonmonotonicity of the entry-deterrence incentive can lead to a nonmonotonic relationship between equilibrium investment levels and market size.Our approach to testing whetherfirms are actively trying to deter entry is a classic reduced-form approach:we identify a prediction of the strategic investment model that dif-fers depending on whetherfirms take entry-deterrence benefits into account when choosing1Some of the classic papers in this literature are Bulow,Geanakoplos,and Klemperer(1985),Fudenberg and Tirole(1983a,1983b,1984),Dixit(1980),Schmalensee(1978,1981),Gelman and Salop(1983),Judd (1985),Aghion and Bolton(1987),and Cooper(1986).their actions;and then test this prediction.The formal results underlying the approach arepresented in Section3.Our main IO theory result is a demonstration that,under specified conditions,actions will be monotonically related to a market size parameter iffirms are not influenced by an entry deterrence motive.Several examples are used to provide intuition for the required conditions.The important implication of the theoretical result is that one can reject the null hypothesis thatfirms are not actively trying to deter entry by testing and rejecting the hypothesis that there is a monotonic relationship betweenfirms’actions and market size in a cross-section of markets.Several recent papers have discussed ways of performing statistical hypothesis tests that a relationship is monotonically increasing.2Section4contains a brief discussion of this lit-erature,a description of the tests we will use,and some additional monotonicity theorems relating to models with measurement error and endogenous right-hand side variables.Ro-bustness to such factors is a potential advantage of an approach focusing on monotonicity.In Section5we turn our attention to the pharmaceutical industry.The pharmaceutical industry is an important industry that has attracted a great deal of attention in policy circles.It also has several features that make it a nice environment in which to study strategic entry deterrence.One of these is that one can obtain a sizable sample of similarly situated incumbents facing a threat of entry by looking at manufacturers of branded drugs whose patent protection is about to expire.Another is that there are several potential tools that incumbents might use to deter entry,and we were able to obtain data on several of the most important:advertising,product proliferation,and pricing.A third is that there is a change in entry conditions within each market—entry is prohibited until a known date.Our approach to testing for strategic entry deterrence only requires a single cross-section,but having data both on actions immediately prior to patent expiration and actions in earlier years when patent expiration was less salient allows us to also implement a difference-in-differences version of our test.Thefirst thing we do in our analysis is to identify a proxy for“market size”and note that our dataset contains sufficient heterogeneity in market size to make it plausible that2These include Hall and Heckman(2000),Ghosal,Sen and van der Vaart(2000),and Gijbels,Hall,Jones and Koch(2000).we couldfind nonmonotonicities.Specifically,revenue received in the U.S.in the years immediately prior to patent expiration can serve as a proxy for market size because it is a strong predictor of whether generic entry will occur.3The lowest-revenue drugs in our dataset are unlikely to ever face generic competition.For high-revenue drugs generic entry is a near certainty.We then examine four incumbent behaviors that might plausibly be involved in an entry deterrence strategy.“Detail advertising”is the practice of sending representatives to doctors’offices to promote a drug via one-on-one conversations.“Journal advertising”is the placement of advertisements for a drug in medical journals.We use the term“presentation proliferation”to refer tofirms’decisions about whether to sell a drug in small or large number of presentations,e.g.should it be offered just in a100mg tablets or should the firm also produce a50mg tablet,a200mg tablet,a gelcap,an oral liquid,etc.Finally,we examine pricing.The literature on“limit pricing”is one of earliest and best known parts of the entry deterrence literature.4We look for evidence that each of these behaviors is influenced by the entry-deterrence motive in two ways.First,we take a pure cross-sectional approach:we look at the relation-ship between each behavior and pre-expiration revenues in the cross-section of drugs and test whether the relationship is nonmonotonic.Wefind some evidence of nonmonotonicity in the journal advertising data.The form of the nonmonotonicity is that journal adver-tising is unusually low for drugs in intermediate-sized markets.This is what one would expect under a strategic-entry deterrence theory:firms in intermediate-sized markets have an incentive to let their market languish to make it less attractive to generic entrants.Second,we examine how incumbents change their behavior as patent expiration nears. Wefind some evidence of a nonmonotonic relationship between behavior changes and mar-ket size in detail advertising and(less strongly)in product proliferation.The changes in detail advertising have a similar pattern to that noted above:it is most likely to be reduced in the intermediate-sized markets.We conclude that there appears to be some evidence of strategic entry-deterrence by 3Grabowski and Vernon(1992),Bae(1997),and Scott Morton(2000)previously reported similar results.4See,for example,Gaskins(1971),Milgrom and Roberts(1982),Fudenberg and Tirole(1986),and Klemperer(1987).pharmaceutical incumbents.More broadly,we hope that our results also suggest thatmonotonicity tests may be a useful way to provide evidence on“strategic investment”theories in industrial organization and otherfields.Our paper can be seen as related to two empirical literatures in industrial organization. First,a number of papers have previously explored pricing,advertising,and entry in the pharmaceutical industry.5Most closely related to our work is Caves,Whinston and Hurwitz (1991),a descriptive study based on thirty drugs with patents expiring between1976and 1987.They look mostly at the average behavior of incumbents before and after expiration and also separate drugs into low and high revenue categories and see if incumbent adver-tising behavior differs.Scott Morton(2000)focuses on the determinants of generic entry in a data set that overlaps substantially with ours.In addition to looking at exogenous market characteristics,she also looks for effects of incumbents’advertising expenditures on the probability of generic entry.Grabowski and Vernon(1992)also study a panel of drugs with expiring patents and focus on post-entry behavior of both incumbents and generic entrants.Ellison and Wolfram(2006)examine pricing as a potentially strategic investment to forestall future regulation.Theyfind that price increases by pharmaceuticalfirms during the Clinton health care reform debate were related to measures offirms’potential losses from drug price regulation.A second literature to which we contribute is the empirical literature on strategic entry deterrence(and entry accommodation).Developing structural tests of whether particular investments are strategic has been seen as difficult.The one paper we are aware of that has attempted this approach is Kadiyali’s(1996)study of the market forfilm.Kadiyali estimates price and advertising elasticites and argues that observed levels of price and advertising by Eastman Kodak were inconsistent with static monopoly profit maximization but consistent with entry deterring behavior.It has been more common to provide indirect evidence that investments are chosen strategically by showing that investments do affect5Some notable papers are,for instance,Masson and Steiner(1985),Comanor(1986),Grabowski and Vernon(1990),Caves,Whinston and Hurwitz(1991),Frank and Salkever(1992),Scherer(1993),Berndt, Griliches and Rosett(1993),Griliches and Cockburn(1994),Berndt,Cockburn and Griliches(1996),Frank and Salkever(1997),Ellison,Cockburn,Griliches and Hausman(1997),Lu and Comanor(1998),Ellison (1998),Scott Morton(1999),and Ellison and Wolfram(2006).There are a number of books of interest about the economics of pharmaceuticals including Walker(1971),Schwartzman(1976),Temin(1980),and Schweitzer(1997).future competition(which will lead us to conclude that investments must be strategic if we believe thatfirms are rational and aware of the effect on competition).One can think of Chevalier’s(1995a,1995b)studies of the effect of capital restructuring on entry and exit and supermarket pricing,Lieberman’s(1987)discussion of the responses by incumbents in chemical industries to rivals’additions of capacity,and Scott Morton’s(2000)discussion of the effects of advertising on entry as providing evidence of this sort.Lieberman also looks for evidence of entry deterring behavior in cross-sectional patterns in examining whether there is more excess capacity in markets which are more concentrated.A third approach taken by Goolsbee and Syverson(2004)in the airline industry is to examine how incumbent behavior changes in response to exogenous changes in potential entry that otherwise have no effect on current competitive conditions.62Strategic entry deterrenceIn this section we develop a simple model to review the idea of strategic entry deterrence and bring out its implications in a framework suited to empirical applications.We use a numerical example to illustrate how nonmonotonic patterns can arise in cross-section data.2.1A modelThe prototypical model of strategic entry deterrence is a three-stage game like thefirst one in Figure1.In thefirst stage,the incumbentfirm1chooses an investment level A at a cost of c(A).Assume that c (A)>0and c (A)≥0.Before the second stage,the potential entrant(firm2)observes the incumbent’s choice of A.Firm2then chooses whether to enter the market,which requires paying a sunk cost of E.In the third stage,either the incumbent is a monopolist or the incumbent and entrant compete as duopolists.If the incumbent is a monopolist,assume that it chooses some action x m1(A)in the third period(A)≡π1(x m1(A),A).If entry occurs,assume that the and as a result earns profits,πm∗1unique Nash equilibrium of the third stage game involves thefirms choosing actions x∗1(A)(A)and and x∗2(A)and receiving profitsπd∗i(A)≡πd i(x∗1(A),x∗2(A),A).Assume thatπm∗1 6Other approaches have also been taken in a few papers.Smiley(1988)reports evidence from surveys offirms about what strategies they use to deter entry.Cooper,Garvin and Kagel(1997)examines a limit-pricing model experimentally.Dafny(2005)applies our approach in studying hospital markets.πd∗i(A)are concave,and that thefirms’best responses are always interior and given by theunique solution to thefirst-order conditions.Tirole(1988)describes how a large number of classic papers in industrial organization (and corporatefinance and international trade)can befit within this framework.The key insight is that the“investment”A can be any action that is not easily undone.The could be a standard investment like building a new plant that will have a lower marginal production cost.It could also,however,be something like a lobbying expenditure,choosing a product design that makes the product less(or more)similar to other products,building up goodwill through an advertising campaign,or taking on debt.Entry deterring investments can be welfare-reducing,but this need not be the case.The one departure we have made from the way strategic investment models are presented in Tirole(1988)(and in most papers)is that we assume that the entry cost E is stochastic with CDF F(E)and that Firm2learns the entry cost before making its entry decision. The primary consequence is that Firm1will not know for sure whether entry will occur when making its investment decision.We think that this is a more realistic depiction of the situationfirms face in the real-world and as well as a necessary modification for empirical applications.2.2The strategic entry deterrence incentiveIn this model,the incumbentfirm1is said to practice“strategic entry deterrence”in that it “distorts”A away from the level that maximizes profits(holdingfirm2’s entry probability fixed)in hopes of deterring entry byfirm2.More precisely,let A∗ED be the sequential equilibrium choice of A in this model.What IO economists mean when they say that investment is“distorted”is that A∗ED differs from the investment level,A∗ND,that would be chosen in the second game pictured in Figure1.7In the second game,firm2does not observefirm1’s investment level until after the entry decision has been made.Hence, investment can not have a causal effect on the entry decision.The nonstrategic investment7To avoid confusing people who know the literature we should note that we have simplified the standard presentation to omit any mention of strategic entry accomodation.Our assumption that A is observed att=212in the model“with no strategic entry deterrence motive”implies that both A∗ND and A∗ED reflectstrategic entry accomodation effects.Hence,any differences between A∗ED and A∗ND are entirely due to entry deterrence motives.Strategic Entry Deterrence Modelt =1t =2t =3Incumbentchooses Aat cost c (A )Potential entrant learns E .Chooses whether toenter at cost E Monopolist chooses x 1or duopolistschoose x 1,x 2Profits:πi (x 1,x 2,A)t =112Potential entrant observes A Investment With No Entry Deterrence Motivet =1t =2t =3Incumbentchooses Aat cost c (A )Potential entrant learns E .Chooses whether toenter at cost E Monopolist chooses x 1or duopolistschoose x 1,x 2Profits:πi (x 1,x 2,A)t =212Potential entrant observes A Figure 1:The modellevel A∗ND can be thought of either as reflecting what would happen if there was no entry deterrence motive or as reflecting what would happen if there was an entry deterrence motive but the incumbent ignored it.Whether incumbents behave in this manner can be of interest for several reasons.For example,antitrust authorities may insist thatfirms do not take actions that serve only to eliminate future competition,and economists may want to know whetherfirms are sufficiently rational and forward looking to invest strategically (and whether the long-run consequences of an investment are what we think they are).Aspects of the strategic entry deterrence motive are most apparent in thefirst-order conditions that describe A∗ED and A∗ND.In the strategic entry deterrence model,firm1’s expected profit is a function of itsfirst period investment:E(π1(A))=F(πd∗2(A))πd∗1(A)+(1−F(πd∗2(A)))πm∗1(A)−c(A).In the model with no strategic entry deterrence motive,firm1’s expected profit depends both on the actual value of A and onfirm2’s belief about the value of A that was chosen in thefirst period.In equilibrium,firm2will assign probablility one tofirm1having chosen A∗ND,sofirm1’s profit function isE(π1(A,A∗ND))=F(πd∗2(A∗ND))πd∗1(A)+(1−F(πd∗2(A∗ND)))πm∗1(A)−c(A).Thefirst-order conditions for the equilibrium investment levels in the two models are thusc (A∗ND)=F(πd∗2(A∗ND))∂πd∗1∂A(A∗ND)+(1−F(πd∗2(A∗ND)))∂πm∗1∂A(A∗ND)c (A∗ED)=F(πd∗2(A∗ED))∂πd∗1∂A(A∗ED)+(1−F(πd∗2(A∗ED)))∂πm∗1∂A(A∗ED)+(πd∗1(A∗ED)−πm∗1(A∗ED))dπd∗2dA(A∗ED)f(πd∗2(A∗ED)).The difference between the twofirst order conditions is the presence of thefinal term in thefirst-order condition for A∗ED.This term is the“strategic entry deterrence”incentive. Becausefirm1’s profit is higher when it is a monopolist,it has an incentive to distort its investment to reducefirm2’s profit(which reduces the likelihood offirm2entering).The main observation about the strategic entry deterrence incentive we would like to highlight is that it may be larger in intermediate-sized markets than in very small or verylarge markets.The incentive is a product of three terms.The third of these,f(πd∗2(A∗ED)), is the likelihood thatfirm2’sfixed entry costs are exactly equal to the equilibrium profits firm2would earn at the post-entry stage,makingfirm2indifferent between entering and not entering.In very small markets this likelihood will be small because thefixed entry costs will almost surely be much larger than the duopoly profits.In very large markets it will be small because thefixed entry costs will almost surely be much smaller than the duopoly profits.In intermediate-sized markets there is a greater chance that the investment will have a pivotal effect on entry.2.3An example of entry-deterrence in a cross-section of marketsIn this section,we present a concrete example of a strategic investment model and discuss cross-sectional implications.Example1Consider a cross-section of markets.Suppose that the i th market has a mass z i of potential consumers,but that the markets are otherwise identical.Let A reflect expen-ditures on a form of advertising that raises potential consumers’valuations for all products in the product class.More specifically,assume that each market contains consumers with heterogeneous types,θ,distributed uniformly uniformly on[0,1],and that if the monopolist spends z i A2/2on advertising in market i,a consumer of typeθreceives utilityθA−p1if he buys the(branded)good fromfirm1at price p1,12θA−p2if he buys the(generic)good fromfirm2at price p2,and zero if he buys neither good.In thefinal period of this model it is easy to check that a monopolist sets p1=A2andreceives profit zA4.The duopoly equilibrium prices are p∗1=27A and p∗2=114A.Duopolyprofits are849zA and149zA.Figure2contains a graph of the equilibrium advertising levels in this model when the distribution F of entry costs is log normal with mean0.0025and variance0.0015.In themodel without entry deterrence motives,A declines smoothly from14at z=0to849in thelimit as z→∞.8When there is also an entry deterrence motive,advertising levels are similar when z is small,but substantially lower in markets of small to intermediate size8Note that in order to show what happens as z goes from zero to infinity we have rescaled the x-axis on the graph using x=z/(z+1).asfirm1distorts its advertising downward to deter entry.In larger marketsfirm1begins to give up on entry deterrence,and the advertising levels in the strategic entry deterrence model approach the equilibrium values of the model without entry deterrence.A notable feature of this example is that the relationship between advertising and market size is monotonic in the model without the entry deterrence incentive and nonmonotonic in the model with the entry deterrence incentive.In the section that follows we discuss the generality of this observation and the possibility of basing tests of strategic intent on it.2.4An aside on structural estimationFew empirical papers have attempted to provide evidence on strategic entry deterrrence via structural methods.Atfirst,one might think that this is surprising:in theory a structural test is as simple as testing which of the competingfirst-order conditions for A bestfits the data.In practice,however,such an approach can be very difficult.By the very nature of these models,entry deterrence can only occur when investments have long term consequences,which is a case when econometric estimates are more difficult.Also, to compute the terms in thefirst order conditions,one needs estimates of the incumbent’s prior on the likelihood of entry,and what the long run benefit of the investment would have been in the counterfactual state of the world in which entry did or did not occur.Each of these can also be a challenge to estimate.93Some Results on Monotonicity and Entry Deterrence Mo-tivesIn the classic reduced-form approach to empirical industrial organization,one identifies where competing models make different predictions and then tests those differences.The null hypothesis for the test we have in mind is that investments are not influenced by the strategic entry deterrence motive.In this section we discuss conditions under which invest-ments that are not influenced by the strategic entry deterrence motive will be monotone in the market size.Under those conditions,if the data are nonmonotone,one can conclude9See Kadiyali(1996)for one structural estimation.Recent advances in the estimation of dynamic struc-tural models,e.g.Aguirregabiria and Mira(2007),Bajari,Benkard,and Levin(2006),and Berry,Ostrovsky, and Pakes(2005),should provide additional opportunities for work along these lines.that investments are influenced by the strategic entry deterrence motive(or that auxilliary assumptions of our propositions are violated).3.1A basic monotonicity result:the direct and competition effects Consider the model of investment without an entry deterrence motivation.Suppose that the profit and cost functions also depend on a characteristic z of the market.Our leading example will be the number of potential consumers in the market.Assume that the variable z is ordered so that larger values of z correspond to markets that are more profitable for firm2,i.e.,∂∂zπd∗2(A,z)>0.In the nonstrategic investment model,investments will covary with z for two reasons.Definition1The“direct effect”of z on A is F(π∗2)∂2πd∗1∂z∂A +(1−F(π∗2))∂2πm∗1∂z∂A−∂2c∂z∂A.The direct effect is positive if increasing z raises the marginal benefit from the investment more than it raises the marginal cost of the investment(holding entry probabilitiesfixed). When the direct effect is positive,it gives the incumbent an incentive to invest more when z is larger.A negative direct effect gives the opposite incentive.Definition2The“competition effect”of z on A is∂πd∗1∂A −∂πm∗1∂A.The competition effect is positive if the marginal benefit of the investment is larger when firm1is engaged in duopoly competition than it is whenfirm1is a monopolist.A larger value of z makes it more likely thatfirm2will enter.When the competition effect is positive,it provides an incentive forfirm1to invest more when z is larger.The following simple proposition identifies a set of circumstances in which investment levels will be monotone in z.Proposition1Let A∗ND(z)be the equilibrium investment level in the model of investment absent entry deterrence motivations described above.Suppose dπd∗2dz>0.10Then A∗ND(z) 10Note that this does involve an additional assumption.We had earlier assumed just that z was orderedso that∂πd∗2∂z >0.Because dπd∗2dz=∂πd∗2∂z+∂πd∗2∂AdAdz,the added assumption can be thought of as a requirementthat the direct effect of z onfirm2’s profits is greater than the indirect effect that comes fromfirm1changing its investment level in response to changing market conditions.While this assumption is often satisfied,itis stronger than is necessary.By expanding dπd∗2dz before solving for dAdzit is easy to see that it suffices tois monotone increasing if the direct and competition effects are positive and A∗ND(z)is monotone decreasing if the direct and competition effects are negative.11The proof of this proposition is given in the Appendix.Remark1:Proposition1is not a result that says that investment without an entry deterrence motive is monotone in z provided some minor technical conclusions hold.We get monotonicity if the direct and competition effects work in the same direction.In some applications,the two effects go in the same direction.In others they would not.One must think about whether an application is of the former or latter type before interpreting a violation of monotonicity as evidence that investments reflect strategic entry deterrence concerns.Remark2:When examining the relationship between investment levels and mar-ket sizes,one has substantial latitude in defining the left-and right-hand-side variables. Monotone transformations of either variable(like taking the log of A)will not affect whether a relationship is monotonic,but other choices one makes in defining variables clearly can. For example,a graph of total advertising expenditure vs.population could look very differ-ent from a graph of per capita advertising expenditure vs.population.Appropriate choices for variable definitions will reflect two concerns:the variables should be chosen so that one would expect the direct and competition effects to be of the same sign;and one should define them so that the direct effects are not so strong so as to make it implausible that strategic entry deterrence motives will be strong enough to lead to nonmonotonicities.For exam-ple,in many applications total advertising expenditures will be approximately proportional instead add the assumption that∂2c ∂A2−F(π∗2)∂2πd∗1∂A2−(1−F(π∗2))∂2πm∗1∂A2>f(π∗2)∂πd∗2∂A∂πd∗1∂A−∂πm∗1∂A.This will always hold if the direction in whichfirm1changes A as competition becomes more likely reduces firm2’s profits(so that the right hand side is negative).For example,this would be the case for an investment in a form of norivalrous advertising which raised consumer awareness of or valuation for all products in a product class.Otherwise,it will be necessary that the term on the right hand side not be too large,which will hold,for example,if the distribution of entry costs is sufficiently diffuse so that the density term is sufficiently small.11To make the propositions easier to read,we have written them using words like increasing and positive rather than nondecreasing and nonnegative.The results extend in all of the obvious ways,e.g.investment is monotone nondecreasing if the direct and competition effects are both nonnegative,and investment is monotone increasing if the direct and competition effects are both nonnegative and one is strictly positive.。
专业英语课文翻译
School of chemical engineering and pharmaceuticaltest tubes 试管test tube holder试管夹test tube brush 试管刷test tube rack试管架beaker烧杯stirring搅拌棒thermometer温度计boiling flask长颈烧瓶Florence flask平底烧瓶flask,round bottom,two-neck boiling flask,three-neck conical flask锥形瓶wide-mouth bottle广口瓶graduated cylinder量筒gas measuring tube气体检测管volumetric flask容量瓶transfer pipette移液管Geiser burette(stopcock)酸式滴定管funnel漏斗Mohr burette(with pinchcock)碱式滴定管watch glass表面皿evaporating dish蒸发皿ground joint磨口连接Petri dish有盖培养皿desiccators干燥皿long-stem funnel长颈漏斗filter funnel过滤漏斗Büchner funnel瓷漏斗separatory funnel分液漏斗Hirsh funnel赫尔什漏斗filter flask 吸滤瓶Thiele melting point tube蒂勒熔点管plastic squeez e bottle塑料洗瓶 medicine dropper药用滴管rubber pipette bulb 吸球microspatula微型压舌板pipet吸量管mortar and pestle研体及研钵filter paper滤纸Bunsen burner煤气灯burette stand滴定管架support ring支撑环ring stand环架distilling head蒸馏头side-arm distillation flask侧臂蒸馏烧瓶air condenser空气冷凝器centrifuge tube离心管fractionating column精(分)馏管Graham condenser蛇形冷凝器crucible坩埚crucible tongs坩埚钳beaker tong烧杯钳economy extension clamp经济扩展夹extension clamp牵引夹utility clamp铁试管夹hose clamp软管夹 burette clamp pinchcock;pinch clamp弹簧夹 screw clamp 螺丝钳ring clamp 环形夹goggles护目镜stopcock活塞wire gauze铁丝网analytical balance分析天平分析化学absolute error绝对误差accuracy准确度assay化验analyte(被)分析物calibration校准constituent成分coefficient of variation变异系数confidence level置信水平detection limit检出限determination测定estimation 估算equivalent point等当点gross error总误差impurity杂质indicator指示剂interference干扰internal standard内标level of significance显着性水平 limit of quantitation定量限masking掩蔽matrix基体precision精确度primary standard原始标准物purity纯度qualitative analysis定性分析 quantitative analysis定量分析random error偶然误差reagent试剂relative error相对误差robustness耐用性sample样品relative standard deviation相对标准偏差 selectivity选择性sensitivity灵敏度specificity专属性titration滴定significant figure有效数字solubility product溶度积standard addition标准加入法standard deviation标准偏差standardization标定法stoichiometric point化学计量点systematic error系统误差有机化学acid anhydride 酸酐acyl halide 酰卤alcohol 醇aldehyde 醛aliphatic 脂肪族的alkene 烯烃alkyne炔allyl烯丙基amide氨基化合物amino acid 氨基酸aromatic compound 芳香烃化合物amine胺butyl 丁基aromatic ring芳环,苯环 branched-chain支链chain链carbonyl羰基carboxyl羧基chelate螯合chiral center手性中心conformers构象copolymer共聚物derivative 衍生物dextrorotatary右旋性的diazotization重氮化作用dichloromethane二氯甲烷ester酯ethyl乙基fatty acid脂肪酸functional group 官能团general formula 通式glycerol 甘油，丙三醇heptyl 庚基heterocyclie 杂环的hexyl 己基homolog 同系物hydrocarbon 烃,碳氢化合物hydrophilic 亲水的hydrophobic 疏水的hydroxide 烃基ketone 酮levorotatory左旋性的methyl 甲基molecular formula分子式monomer单体octyl辛基open chain开链optical activity旋光性（度）organic 有机的organic chemistry 有机化学organic compounds有机化合物pentyl戊基phenol苯酚phenyl苯基polymer 聚合物，聚合体propyl丙基ring-shaped环状结构 zwitterion兼性离子saturated compound饱和化合物side chain侧链straight chain 直链tautomer互变（异构）体structural formula结构式triglyceride甘油三酸脂unsaturated compound不饱和化合物物理化学activation energy活化能adiabat绝热线amplitude振幅collision theory碰撞理论empirical temperature假定温度enthalpy焓enthalpy of combustion燃烧焓enthalpy of fusion熔化热enthalpy of hydration水合热enthalpy of reaction反应热enthalpy o f sublimation升华热enthalpy of vaporization汽化热entropy熵first law热力学第一定律first order reaction一级反应free energy自由能Hess’s law盖斯定律Gibbs free energy offormation吉布斯生成能heat capacity热容internal energy内能isobar等压线isochore等容线isotherm等温线kinetic energy动能latent heat潜能Planck’s constant普朗克常数potential energy势能quantum量子quantum mechanics量子力学rate law速率定律specific heat比热spontaneous自发的standard enthalpy change标准焓变standard entropy of reaction标准反应熵standard molar entropy标准摩尔熵standard pressure标压state function状态函数thermal energy热能thermochemical equation热化学方程式thermodynamic equilibrium热力学平衡uncertainty principle测不准定理zero order reaction零级反应 zero point energy零点能课文词汇实验安全及记录：eye wash眼药水first-aid kit急救箱gas line输气管safety shower紧急冲淋房water faucet水龙头flow chart流程图loose leaf活页单元操作分类：heat transfer传热Liquid-liquid extraction液液萃取liquid-solid leaching过滤vapor pressure蒸气压membrane separation薄膜分离空气污染：carbon dioxide 二氧化碳carbon monoxide一氧化碳particulate matter颗粒物质photochemical smog光化烟雾primary pollutants一次污染物secondary pollutants二次污染物 stratospheric ozone depletion平流层臭氧消耗sulfur dioxide二氧化硫volcanic eruption火山爆发食品化学：amino acid氨基酸,胺amino group氨基empirical formula实验式,经验式fatty acid脂肪酸peptide bonds肽键polyphenol oxidase 多酚氧化酶salivary amylase唾液淀粉酶 steroid hormone甾类激素table sugar蔗糖triacylglycerol三酰甘油,甘油三酯食品添加剂：acesulfame-K乙酰磺胺酸钾,一种甜味剂adrenal gland肾上腺ionizing radiation致电离辐射food additives食品添加剂monosodium glutamate味精,谷氨酸一钠（味精的化学成分）natural flavors天然食用香料,天然食用调料nutrasweet天冬甜素potassium bromide 溴化钾propyl gallate没食子酸丙酯sodium chloride氯化钠sodium nitraten硝酸钠sodium nitrite亚硝酸钠trans fats反式脂肪genetic food转基因食品food poisoning 食物中毒hazard analysis and critical control points (HACCP)危害分析关键控制点技术maternal and child health care妇幼保健护理national patriotic health campaign committee(NPHCC) 全国爱国卫生运动委员会rural health农村卫生管理the state food and drug administration (SFDA)国家食品药品监督管理局光谱：Astronomical Spectroscopy天文光谱学Laser Spectroscopy激光光谱学 Mass Spectrometry质谱Atomic Absorption Spectroscopy原子吸收光谱Attenuated T otal Reflectance Spectroscopy衰减全反射光谱Electron Paramagnetic Spectroscopy电子顺磁谱Electron Spectroscopy电子光谱Infrared Spectroscopy红外光谱Fourier Transform Spectrosopy傅里叶变换光谱Gamma-ray Spectroscopy伽玛射线光谱Multiplex or Frequency-Modulated Spectroscopy复用或频率调制光谱X-ray SpectroscopyX射线光谱色谱：Gas Chromatography气相色谱High Performance Liquid Chromatography高效液相色谱Thin-Layer Chromatography薄层色谱magnesium silicate gel硅酸镁凝胶retention time保留时间mobile phase流动相stationary phase固定相反应类型：agitated tank搅拌槽catalytic reactor催化反应器batch stirred tank reactor间歇搅拌反应釜continuous stirred tank 连续搅拌釜exothermic reactions放热反应pilot plant试验工厂fluidized bed Reactor流动床反应釜multiphase chemical reactions 多相化学反应packed bed reactor填充床反应器redox reaction氧化还原反应reductant-oxidant氧化还原剂acid base reaction酸碱反应additionreaction加成反应chemical equation化学方程式valence electron价电子combination reaction化合反应hybrid orbital 杂化轨道decomposition reaction分解反应substitution reaction取代(置换)反应Lesson5 Classification of Unit Operations单元操作Fluid flow流体流动它涉及的原理是确定任一流体从一个点到另一个点的流动和输送。
熵值法 excel 计算过程
熵值法 excel 计算过程1.打开Excel软件，新建一个表格。
Open Excel software and create a new spreadsheet.2.在第一列依次输入各个选项的名称。
Enter the names of the options in the first column.3.在接下来的列中，依次输入各个选项的相关数据。
Enter the relevant data for each option in the following columns.4.在Excel中选择一个空白单元格，输入以下函数：=ENTROPY （A2:A52）Select a blank cell in Excel and enter the following function: =ENTROPY(A2:A52)5.按下回车键，Excel会自动计算出这些选项的熵值。
Press Enter, and Excel will automatically calculate the entropy of the options.6.熵值法是一种用来度量不确定性的方法。
The entropy method is a way to measure uncertainty.7.它可以帮助我们对各个选项之间的差异进行量化评估。
It can help us quantitatively evaluate the differences between the options.8.在excel中使用熵值法可以方便快捷地进行大量数据的计算。
Using the entropy method in Excel can quickly and easily calculate large amounts of data.9.通过比较不同选项的熵值，我们可以找出最优的选择。
By comparing the entropy of different options, we can find the optimal choice.10.熵值法在决策分析和风险评估中有着广泛的应用。
stata熵权法计算指令
stata熵权法计算指令英文回答：The entropy weight method (EWM) is a widely used objective weighting method in the field of decision-making. It utilizes the concept of information entropy to determine the weights of different criteria, ensuring that the weights are unbiased and reflect the relative importance of each criterion.The EWM algorithm involves the following steps:1. Normalize the decision matrix to ensure that all criteria are on the same scale.2. Calculate the entropy value for each criterion using the following formula:E_j = ∑(p_ij log(p_ij))。
where p_ij is the normalized value of the jthcriterion for the ith alternative.3. Calculate the weight for each criterion using the following formula:W_j = (1 E_j) / ∑(1 E_k)。
where k represents all criteria.The EWM has several advantages. It is objective anddoes not require subjective judgments from decision-makers. It also considers the uncertainty and variation in the data, ensuring that the weights are robust and reliable.However, the EWM also has some limitations. It assumes that all criteria are equally important, which may not always be the case in real-world decision-making scenarios. Additionally, the EWM can be sensitive to outliers in the data, which can affect the calculated weights.Despite these limitations, the EWM remains a valuabletool for decision-making, especially when the criteria are complex and uncertain. It provides a systematic andunbiased approach to determining the weights of different criteria, leading to more informed and defensible decisions.中文回答：熵权法。
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Z. Wahrscheinlichkeitstheorieverw. Gebiete 58, 69-85 (1981)
Wahrscheinlichkeitstheorie
und verwandteGebiete 9 Springer-Vertag 1981
Almost Sure Entropy and the Variational Principle for Random Fields with Unbounded State Space
(1.4)
~v(X)= ~ (-1)lvl-lWlUw(x).
W=V
It follows from (1.2) and (1and
WcV
@v(X)=0 if xz=0 for some ieV.
Usually we will work rather with the interaction than with the potential.
1. Definitions and Assumptions
We consider the configuration space X=(IR~) Zd, i.e. the set of all functions x: 2U~IR ~. x~ will be the value at the site i. Let ~ be the a-field generated by the projections x~x~, is2U. A lR"-valued random field is then a probability measure on (X, ~). A random field # will be called stationary if # is invariant under all shifttransformations 0~, i~TZ where (Oix)j=xj+ i. We will call # d, tempered (in the sense of Ruelle [11]) if (1.1)
0044-3719/81/0058/0069/$03.40
70
H. Kiinsch
equal difference of free energies holds also for unbounded state space. This formula makes it possible to prove Prestons result about information gain mentionned before also in cases of infinite range of the energy. Moreover our approach gives new proofs for the existence of pressure and of Pirlots results. It consists essentially in proving that the conditional density approximates in some sense the absolute density (Sect. 3). In the case of finite state space this is not very hard to prove (FSllmer [-1l, formula (4.22)), but in our case it requires rather complicated estimates and a condition somewhat stronger than regularity. In Sect. 2 we prove the almost sure and Ll-Convergence of the energy with a version of the ergodic theorem for nonadditive functions. Section 4 contains the main results. The author wishes to express his gratitude to H. FSllmer for suggesting the problem and helping with valuable suggestions. Thanks are also due to Gallavotti for a stimulating discussion.
Uv(x)= Uw(x)
if W _ V and xi=O for all i t V\W,
U~(0)=0.
The corresponding interaction I is defined by (1.3)
Iv, w(X)= Uww(X ) - Uv(x ) - Uw(X)
(V~ W= r
and the potential ~ by
given the values Xj, jr can be written in terms of an energy function for all finite V~_7/d (cf. e.g. [8]). If several Gibbsian fields with the same energy exist, it is interpreted as phase transition. The Gibbs variational principle says that among all stationary random fields the Gibbsian fields are characterized by the fact that they minimize a suitable free energy which is a justification for calling the Gibbsian fields equilibria. It was proved by Lanford and Ruelle [-4] if the Xi take only the two values 0 or 1. F611mer [-1] discovered a connection of this with information theoretic quantities: He proved that the information gain of a stationary field over a Gibbsian field is the difference of the free energies. The variational principle is then equivalent to the claim that two stationary fields have information gain zero iff they have the same conditional distributions for finite sets given the outside. If the state space is unbounded, new difficulties arise because in general uniform bounds are no more available. Usually one works here with the superstability and regularity assumptions (see [11, 12]) which give powerful estimates for the conditional distributions (Ruelle [-12]). With the help of these estimates Lebowitz and Presutti [-5] proved the existence of the so called pressure, and with this result Pirlot [-7] showed that Gibbsian fields minimize the free energy and that two Gibbsian fields with the same energy have information gain zero. A general result of Preston [8] says in our situation that if the information gain of a stationary random field with respect to a Gibbsian field with energy of finite range is zero, it must be Gibbsian with the same energy. The case of infinite range is still open. This paper contains the following results: We prove almost sure and L 1convergence of the entropy for Gibbsian fields (d-dimensional version of the theorem of McMillan-Breiman) which gives us also an almost sure version of the variational principle. Secondly we show that the formula information gain
Almost Sure Entropy and the Variational Principle We will always suppose that the energy is (1.6)
#{U ~ ( Z x~<Ne(2n+l)a)} =1,